Symmetric semigroups of integers generated by 4 elements

Abstract

Let K be an arbitrary field, t transcendental over over K. It is shown that if the numerical semigroup of nonnegative integers, generated nonredundantly by 4 elements, is symmetric, then the prime ideal of all polynomials f(x1,x2,X3,x4) e K[x1,x2,x3,x4] such that\(f(t^{n_1 } ,t^{n_2 } ,t^{n_3 } ,t^{n_4 } ) = 0\) is generated by 3 or 5 elements. From this, arithmetic conditions for the generators are obtained.